What is convergence in machine learning?
When formulating a problem in deep learning, we need to come up with a loss function, which uses model weights as parameters. Back-propagation starts at an arbitrary point on the error manifold defined by the loss function and with every iteration intends to move closer to a point that minimises error value by updating the weights. Essentially for every possible set of weights the model can have, there is an associated loss for a given loss function, with our goal being to find the minimum point on this manifold. 

Convergence is a term mathematically most common in the study of series and sequences. A model is said to converge when the series $s(n) = loss_{w_n}(\hat y, y)$ (Where $w_n$ is the set of weights after the $n$'th iteration of back-propagation and $s(n)$ is the $n$'th term of the series) is a converging series. The series is of course an infinite series only if you assume that loss = 0 is never actually achieved, and that learning rate keeps getting smaller. 

Essentially meaning, a model converges when its loss actually moves towards a minima (local or global) with a decreasing trend. Its quite rare to actually come across a strictly converging model but convergence is commonly used in a similar manner as convexity is. Strictly speaking rarely exists practically, but is spoken in a manner telling us how close the model is to the ideal scenario for convexity, or in this case convergence.